Postgres 15

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CREATE TABLE end_user (id int, profile jsonb);
INSERT INTO end_user VALUES
  (1, '{"first_segment_exited_at": "2023-05-22T03:09:37.825Z"}')
, (2, '{"first_segment_exited_at": "2023-05-22T03:09:37Z"}')  -- no fractional sec
, (3, '{"first_segment_exited_at": "2023-05-22T03:09:37"}')   -- AND no trailing "Z"
;

CREATE TABLE

INSERT 0 3

SELECT *
FROM   end_user
WHERE  profile @? '$.first_segment_exited_at ? (@.datetime() < "2023-06-22T03:09:37".datetime())'

id
profile

3
{"first_segment_exited_at": "2023-05-22T03:09:37"}

SELECT 1

-- filter can use just date like you did, too
SELECT id, profile
FROM   end_user
WHERE  profile @? '$.first_segment_exited_at ? (@.datetime() < "2023-05-25".datetime())'::jsonpath;

id
profile

3
{"first_segment_exited_at": "2023-05-22T03:09:37"}

SELECT 1

-- but the format must be valid!
SELECT *
FROM   end_user
WHERE  profile @? '$.first_segment_exited_at ? (@.datetime() < "2023-06-22T03:09:37Z".datetime())';

id
profile

SELECT 0